3.1600 \(\int \frac {(a+\frac {b}{x})^8}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac {\left (a+\frac {b}{x}\right )^9}{9 b} \]

[Out]

-1/9*(a+b/x)^9/b

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \[ -\frac {\left (a+\frac {b}{x}\right )^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^2,x]

[Out]

-(a + b/x)^9/(9*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^8}{x^2} \, dx &=-\frac {\left (a+\frac {b}{x}\right )^9}{9 b}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 96, normalized size = 6.00 \[ -\frac {a^8}{x}-\frac {4 a^7 b}{x^2}-\frac {28 a^6 b^2}{3 x^3}-\frac {14 a^5 b^3}{x^4}-\frac {14 a^4 b^4}{x^5}-\frac {28 a^3 b^5}{3 x^6}-\frac {4 a^2 b^6}{x^7}-\frac {a b^7}{x^8}-\frac {b^8}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^2,x]

[Out]

-1/9*b^8/x^9 - (a*b^7)/x^8 - (4*a^2*b^6)/x^7 - (28*a^3*b^5)/(3*x^6) - (14*a^4*b^4)/x^5 - (14*a^5*b^3)/x^4 - (2
8*a^6*b^2)/(3*x^3) - (4*a^7*b)/x^2 - a^8/x

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fricas [B]  time = 0.99, size = 88, normalized size = 5.50 \[ -\frac {9 \, a^{8} x^{8} + 36 \, a^{7} b x^{7} + 84 \, a^{6} b^{2} x^{6} + 126 \, a^{5} b^{3} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{3} b^{5} x^{3} + 36 \, a^{2} b^{6} x^{2} + 9 \, a b^{7} x + b^{8}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^2,x, algorithm="fricas")

[Out]

-1/9*(9*a^8*x^8 + 36*a^7*b*x^7 + 84*a^6*b^2*x^6 + 126*a^5*b^3*x^5 + 126*a^4*b^4*x^4 + 84*a^3*b^5*x^3 + 36*a^2*
b^6*x^2 + 9*a*b^7*x + b^8)/x^9

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giac [A]  time = 0.15, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x}\right )}^{9}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^2,x, algorithm="giac")

[Out]

-1/9*(a + b/x)^9/b

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maple [B]  time = 0.01, size = 91, normalized size = 5.69 \[ -\frac {a^{8}}{x}-\frac {4 a^{7} b}{x^{2}}-\frac {28 a^{6} b^{2}}{3 x^{3}}-\frac {14 a^{5} b^{3}}{x^{4}}-\frac {14 a^{4} b^{4}}{x^{5}}-\frac {28 a^{3} b^{5}}{3 x^{6}}-\frac {4 a^{2} b^{6}}{x^{7}}-\frac {a \,b^{7}}{x^{8}}-\frac {b^{8}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8/x^2,x)

[Out]

-14*a^4*b^4/x^5-14*a^5*b^3/x^4-28/3*a^6*b^2/x^3-a*b^7/x^8-4*a^7*b/x^2-1/9*b^8/x^9-4*a^2*b^6/x^7-28/3*a^3*b^5/x
^6-a^8/x

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maxima [A]  time = 1.10, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x}\right )}^{9}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^2,x, algorithm="maxima")

[Out]

-1/9*(a + b/x)^9/b

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mupad [B]  time = 1.10, size = 88, normalized size = 5.50 \[ -\frac {a^8\,x^8+4\,a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{3}+14\,a^5\,b^3\,x^5+14\,a^4\,b^4\,x^4+\frac {28\,a^3\,b^5\,x^3}{3}+4\,a^2\,b^6\,x^2+a\,b^7\,x+\frac {b^8}{9}}{x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^8/x^2,x)

[Out]

-(b^8/9 + a^8*x^8 + 4*a^7*b*x^7 + 4*a^2*b^6*x^2 + (28*a^3*b^5*x^3)/3 + 14*a^4*b^4*x^4 + 14*a^5*b^3*x^5 + (28*a
^6*b^2*x^6)/3 + a*b^7*x)/x^9

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sympy [B]  time = 0.69, size = 95, normalized size = 5.94 \[ \frac {- 9 a^{8} x^{8} - 36 a^{7} b x^{7} - 84 a^{6} b^{2} x^{6} - 126 a^{5} b^{3} x^{5} - 126 a^{4} b^{4} x^{4} - 84 a^{3} b^{5} x^{3} - 36 a^{2} b^{6} x^{2} - 9 a b^{7} x - b^{8}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8/x**2,x)

[Out]

(-9*a**8*x**8 - 36*a**7*b*x**7 - 84*a**6*b**2*x**6 - 126*a**5*b**3*x**5 - 126*a**4*b**4*x**4 - 84*a**3*b**5*x*
*3 - 36*a**2*b**6*x**2 - 9*a*b**7*x - b**8)/(9*x**9)

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